EXTENDING TOPOLOGICAL GROUP ACTIONS TO CONFORMAL GROUP ACTIONS

Abstract

A consequence of the resolution of the Nielsen Realization Problem is that if $G$ is any finite topological group of automorphisms of a compact oriented surface $S$, then there exists a complex structure on $S$ so that the action of $G$ extends to a conformal action on $S$. We show that this complex structure is unique if and only if $G$ is a triangle group, and for all other groups, there are infinitely many such actions.